扩散过程在若干交叉领域中的理论及应用
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摘要
扩散过程起源于物理学,之后引起数学学者们的广泛关注,一直以来是随机分析方向的前沿和热点。如何延续这种扩散过程和其它领域的有机结合,体现交叉效用,拓展扩散过程理论的研究视野,是一个具有理论意义和实际应用价值的研究领域。
全文的创新性工作如下：
在第3章中我们研究了扩散过程与随机序理论的交叉。利用扩散过程和条件期望的性质,直接证明了扩散过程随机序的比较定理,包括强序、增凸序、增凹序及Laplace-Stieltjes转移序。然后将随机序方法应用到扩散过程的Fokker-Planck方程中,证实了一类偏微分方程解的弱比较定理。
在第4章中我们探讨了扩散过程与灰色系统、统计推断的交叉。一方面针对小样本观测值的情形,首次提出了用灰估计的方法来估计随机微分方程中的参数,两者间的桥梁是Ito公式。通过实例表明：与原有的方法相较,参数估计的效果相差不大,但我们所需要的数据信息少,在实际中可操作性强,试验成本低。另一方面给出了一类广义的扩散过程(McKean-Vlasov随机微分方程)的参数估计方法,在一些假设条件下,通过一定的变换,将非时齐的Markov过程变形为时齐的Markov过程。再通过对后者作参数估计进而讨论原方程的参数估计问题,得到了参数的极大似然估计和贝叶斯估计,并证明了它们的性质。
在第5章中我们讨论了扩散过程与三个应用领域的交叉。首先研究了非高斯噪声对热盐环流的影响,根据噪声的性质将其用一个α-稳定的levy过程描述,建立了随机Stommel模型,再用数值的方法模拟随着α的变动,方程解的平均逃逸时的变化情况,从而得到盐度差的稳定范围。其次,分析了纳米药物进入机体后的吸附过程,引入白噪声项描述随机因素对该过程的影响,建立了纳米药物的吸附动力学模型,并分析了参数的变动对整个吸附过程的影响,用于确定纳米药物的制备参数及指导用药量。最后,针对离子在运动过程中由于自身带电荷,相互间产生电磁场不再做自由扩散这一情况,引入McKean-Vlasov随机微分方程描述离子的运动状况。推导出转移密度满足的非线性Fokker-Planck方程,用数值计算的方法模拟离子运动的轨道变化。
Diffusion process was origin of the study of diffusion phenomenon in physics, and then was attracted attention by mathematical scholars. It has been the frontier and hot on direction of stochastic analysis. How to keep this organic combination of diffusion process and other fields, reflect cross utility and expand the research horizon of diffusion process theory, is a study field which has theoretical significance and practical application value.
The main innovative results of this paper are as follows.
We discuss stochastic ordering problems of diffusion process in the third chap-ter. Focused on comparing the magnitude and variability of diffusion processes, some properties are intensively demonstrated about strong ordering, increasing convex or-dering, increasing concave ordering and Laplace-Stieltjes transform ordering for d-iffusion processes which are defined as stochastic differential equations respectively. Finally the stochastic ordering was applied to diffusion processes which densities sat-isfied Fokker-Planck equations and the weak comparison theory of partial differential equation is obtained through stochastic ordering method.
In the fourth chapter parameter estimation problem of diffusion processes is ana-lyzed from two aspects. On one hand, grey estimation method is borrowed to discuss parameter estimation of stochastic differential equations driven by Brownian motion for the situation of little information and small sample observations. We evolve a whitenization equation through substituting random variable, then corresponding grey differential equation can be expressed which suited for describing few data and un-certainty condition. On other hand, we research on parameter estimation of a class of generalized diffusion process named McKean-Vlasov stochastic differential equation, containing Maximum Likelihood Estimator and Bayes Estimator.
Applications of diffusion process on three domains are came true in the fifth chapter. Firstly, the effects of non-Gaussian noise on the Stommel box model for the Thermohaline Circulation are considered. The noise is represented by a non-Gaussian α-stable Levy motion with0<α<2. The a value may be regarded as the index of non-Gaussianity. Dynamical features of this stochastic model is examined by computing the mean exit time for various a values. The mean exit time is simulated by numerically solving a deterministic differential equation with nonlocal interac-tions. Secondly, the mechanics of nanoparticles'adhesion and cellular uptake is described in detail, and a stochastic adhesion kinetic model is built by introducing white noise term. Furthermore, using simulation method diagrams are showed the effects to the whole adhesion process due to parameters fluctuations. Lastly since ion has electric charge which create interactions, it is no longer free diffusion. We use McKean-Vlasov stochastic differential equation to delineate ion motion, induce Fokker-Planck equation satisfied by translated density, and simulate varied trajectory of ion motion.
全文的创新性工作如下：
在第3章中我们研究了扩散过程与随机序理论的交叉。利用扩散过程和条件期望的性质,直接证明了扩散过程随机序的比较定理,包括强序、增凸序、增凹序及Laplace-Stieltjes转移序。然后将随机序方法应用到扩散过程的Fokker-Planck方程中,证实了一类偏微分方程解的弱比较定理。
在第4章中我们探讨了扩散过程与灰色系统、统计推断的交叉。一方面针对小样本观测值的情形,首次提出了用灰估计的方法来估计随机微分方程中的参数,两者间的桥梁是Ito公式。通过实例表明：与原有的方法相较,参数估计的效果相差不大,但我们所需要的数据信息少,在实际中可操作性强,试验成本低。另一方面给出了一类广义的扩散过程(McKean-Vlasov随机微分方程)的参数估计方法,在一些假设条件下,通过一定的变换,将非时齐的Markov过程变形为时齐的Markov过程。再通过对后者作参数估计进而讨论原方程的参数估计问题,得到了参数的极大似然估计和贝叶斯估计,并证明了它们的性质。
在第5章中我们讨论了扩散过程与三个应用领域的交叉。首先研究了非高斯噪声对热盐环流的影响,根据噪声的性质将其用一个α-稳定的levy过程描述,建立了随机Stommel模型,再用数值的方法模拟随着α的变动,方程解的平均逃逸时的变化情况,从而得到盐度差的稳定范围。其次,分析了纳米药物进入机体后的吸附过程,引入白噪声项描述随机因素对该过程的影响,建立了纳米药物的吸附动力学模型,并分析了参数的变动对整个吸附过程的影响,用于确定纳米药物的制备参数及指导用药量。最后,针对离子在运动过程中由于自身带电荷,相互间产生电磁场不再做自由扩散这一情况,引入McKean-Vlasov随机微分方程描述离子的运动状况。推导出转移密度满足的非线性Fokker-Planck方程,用数值计算的方法模拟离子运动的轨道变化。
Diffusion process was origin of the study of diffusion phenomenon in physics, and then was attracted attention by mathematical scholars. It has been the frontier and hot on direction of stochastic analysis. How to keep this organic combination of diffusion process and other fields, reflect cross utility and expand the research horizon of diffusion process theory, is a study field which has theoretical significance and practical application value.
The main innovative results of this paper are as follows.
We discuss stochastic ordering problems of diffusion process in the third chap-ter. Focused on comparing the magnitude and variability of diffusion processes, some properties are intensively demonstrated about strong ordering, increasing convex or-dering, increasing concave ordering and Laplace-Stieltjes transform ordering for d-iffusion processes which are defined as stochastic differential equations respectively. Finally the stochastic ordering was applied to diffusion processes which densities sat-isfied Fokker-Planck equations and the weak comparison theory of partial differential equation is obtained through stochastic ordering method.
In the fourth chapter parameter estimation problem of diffusion processes is ana-lyzed from two aspects. On one hand, grey estimation method is borrowed to discuss parameter estimation of stochastic differential equations driven by Brownian motion for the situation of little information and small sample observations. We evolve a whitenization equation through substituting random variable, then corresponding grey differential equation can be expressed which suited for describing few data and un-certainty condition. On other hand, we research on parameter estimation of a class of generalized diffusion process named McKean-Vlasov stochastic differential equation, containing Maximum Likelihood Estimator and Bayes Estimator.
Applications of diffusion process on three domains are came true in the fifth chapter. Firstly, the effects of non-Gaussian noise on the Stommel box model for the Thermohaline Circulation are considered. The noise is represented by a non-Gaussian α-stable Levy motion with0<α<2. The a value may be regarded as the index of non-Gaussianity. Dynamical features of this stochastic model is examined by computing the mean exit time for various a values. The mean exit time is simulated by numerically solving a deterministic differential equation with nonlocal interac-tions. Secondly, the mechanics of nanoparticles'adhesion and cellular uptake is described in detail, and a stochastic adhesion kinetic model is built by introducing white noise term. Furthermore, using simulation method diagrams are showed the effects to the whole adhesion process due to parameters fluctuations. Lastly since ion has electric charge which create interactions, it is no longer free diffusion. We use McKean-Vlasov stochastic differential equation to delineate ion motion, induce Fokker-Planck equation satisfied by translated density, and simulate varied trajectory of ion motion.
引文
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[8]Stefano M.I., Simulation and inference for Stochastic Differential Equations, Springer-Verlag, New York,2007.
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